The basic halachah is that when the congregation/community starts Shabbos (Sabbath) early an individual is forbidden to perform melachah (labor) even though Shabbos has not technically begun.

What is the basis for the congregation to have two minyanim, an early one and a late one? Why doesn't this cause a halachic problem (at least according to the opinions who permit it, not that I know of any who forbid it) if the majority of the kehillah starts early?

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They are only bound by the concept of Al Tifrosh Min HaTzibbur since the tzibbur is keeping a minhag that is based on a mitzvah (tosefes Shabbos or to prevent milachah). Once there is more than one minhag in town/shul the problem is gone. Even if the tzibbur only starts early, if they only do so in the summer and only to eat at a normal time and not for tosefes Shabbos or to prevent milachah, it is not so clear that one should be forced by the tzibbur to receive Shabbos early. (Igros Moshe O.C. III Siman 38)